Excited-State Quantum Phase Transitions

• Excited-state quantum phase transitions are nonthermal singularities in the excited spectra of finite quantum systems, occurring at critical energies associated with stationary points of the Hamiltonian.
• They manifest through density-of-states anomalies, abrupt eigenstate reorganization, and distinct nonequilibrium dynamics such as critical slowing down and anomalous relaxation.
• Experimental platforms like trapped ions, spinor condensates, and cavity QED provide practical insights into ESQPT behavior and finite-size scaling effects.

Excited-state quantum phase transitions (ESQPTs) are nonthermal singularities of the excited spectrum of isolated quantum many-body systems with a finite number of collective degrees of freedom. In contrast to ground-state quantum phase transitions, which are driven by nonanalytic changes of the lowest eigenstate as a control parameter varies, ESQPTs occur at critical energies $E_c$ within the spectrum and are tied semiclassically to stationary points and separatrices of the classical Hamiltonian. Their standard manifestations include singularities in the smoothed density of states, abrupt changes in eigenstate structure, anomalous expectation values, altered degeneracy patterns, and distinctive nonequilibrium dynamics such as critical slowing down, anomalous relaxation, and phase-dependent long-time behavior (Cejnar et al., 2020).

1. Semiclassical definition and general classification

The standard semiclassical formulation expresses the smooth level density as a phase-space volume on the constant-energy shell. For a system with $f$ collective degrees of freedom and classical Hamiltonian $H(x)$, the Weyl component of the density of states is

$$\overline{\rho}(E)=\frac{1}{(2\pi\hbar)^f}\int d^{2f}x\ \delta\!\big(E-H(x)\big),$$

or equivalently as the derivative of the accessible phase-space volume with respect to energy. ESQPTs arise when the energy shell crosses a stationary point $x_c$ satisfying $\nabla H(x_c)=0$, so that the topology of the classical invariant manifolds changes at $E_c=H(x_c)$ (Cejnar et al., 2020).

For nondegenerate stationary points, the singularity is controlled by the number of effective degrees of freedom and by the Morse index $r$, i.e. the number of negative eigenvalues of the Hessian. In the generic classification reviewed for finite-$f$ systems, the $(f-1)$st derivative of $f$0 exhibits either a step discontinuity or a logarithmic divergence, depending on parity properties of the stationary point. This yields the familiar logarithmic divergence of $f$1 itself in effective one-degree-of-freedom systems, while for $f$2 the singularity is shifted to $f$3 (Cejnar et al., 2020).

Interacting boson systems with $f$4 show that this classification remains valid but acquires two complications: the classical Hamiltonian need not decompose trivially into kinetic and potential parts, and the phase space is bounded because of number conservation. As a result, additional ESQPT signatures can originate from nontrivial kinetic stationary points and from phase-space boundaries, not only from extrema of an effective potential (1903.16551).

The constrained-system formulation extends the same logic to Hamiltonians restricted by integrals of motion. Introducing Lagrange multipliers, one identifies stationary points of the constrained dynamics directly from

$f$5

with $f$6 the constraints. The effective number of degrees of freedom is reduced accordingly, and the order of the singular derivative of $f$7 shifts with the reduced $f$8. This extension is essential in algebraic boson models with conserved excitation numbers and in integrable subsectors where fixing additional integrals qualitatively changes the ESQPT pattern (Novotný et al., 2024).

2. Canonical realizations and model classes

A broad range of collective models realizes ESQPTs through the same semiclassical mechanism while differing in algebraic structure, symmetry content, and accessible diagnostics.

Model class	Critical structure	Hallmark signature
Two-level pairing models with $f$9 algebra	$H(x)$0, $H(x)$1	Level clustering and basis localization (Santos et al., 2015)
Infinite-range Ising / LMG	$H(x)$2, same separatrix formula	DOS peak, $H(x)$3 bifurcation, slow evolution (Santos et al., 2016)
Dicke and Tavis–Cummings models	Static ESQPT at $H(x)$4; dynamic ESQPT at $H(x)$5 in the superradiant phase	Discontinuity or logarithmic singularity in DoS (Bastarrachea-Magnani et al., 2013)
Spin-1 spinor Bose gas	ESQPT line $H(x)$6 for $H(x)$7	Topology change of mean-field trajectories (Feldmann et al., 2020)
Raman-dressed spin-orbit-coupled Bose gas	Excited-state boundaries at $H(x)$8 in the $H(x)$9 plane	Excited-stripe phase and stripe-contrast order parameter (Cabedo et al., 2021)
Three-level LMG	Multiple separatrices $$\overline{\rho}(E)=\frac{1}{(2\pi\hbar)^f}\int d^{2f}x\ \delta\!\big(E-H(x)\big),$$0, including $$\overline{\rho}(E)=\frac{1}{(2\pi\hbar)^f}\int d^{2f}x\ \delta\!\big(E-H(x)\big),$$1–$$\overline{\rho}(E)=\frac{1}{(2\pi\hbar)^f}\int d^{2f}x\ \delta\!\big(E-H(x)\big),$$2	ESQPT signatures intertwined with chaos (Mayorgas et al., 26 Feb 2026)

In two-level bosonic models with $$\overline{\rho}(E)=\frac{1}{(2\pi\hbar)^f}\int d^{2f}x\ \delta\!\big(E-H(x)\big),$$3 dynamical algebra, the Hamiltonian

$$\overline{\rho}(E)=\frac{1}{(2\pi\hbar)^f}\int d^{2f}x\ \delta\!\big(E-H(x)\big),$$4

interpolates between spherical and deformed dynamical symmetries. For $$\overline{\rho}(E)=\frac{1}{(2\pi\hbar)^f}\int d^{2f}x\ \delta\!\big(E-H(x)\big),$$5, a separatrix appears at

$$\overline{\rho}(E)=\frac{1}{(2\pi\hbar)^f}\int d^{2f}x\ \delta\!\big(E-H(x)\big),$$6

and the density of states develops a pronounced peak that sharpens with $$\overline{\rho}(E)=\frac{1}{(2\pi\hbar)^f}\int d^{2f}x\ \delta\!\big(E-H(x)\big),$$7 (Santos et al., 2015).

The infinite-range Ising realization of the Lipkin–Meshkov–Glick model,

$$\overline{\rho}(E)=\frac{1}{(2\pi\hbar)^f}\int d^{2f}x\ \delta\!\big(E-H(x)\big),$$8

provides the corresponding $$\overline{\rho}(E)=\frac{1}{(2\pi\hbar)^f}\int d^{2f}x\ \delta\!\big(E-H(x)\big),$$9 case. Its classical potential becomes a double well for $x_c$0, and the ESQPT occurs at the barrier top, producing a local divergence in the density of states and a bifurcation of magnetization branches (Santos et al., 2016).

In atom–field systems, the Dicke and Tavis–Cummings Hamiltonians exhibit two distinct ESQPTs. The semiclassical density of states reveals a static ESQPT at $x_c$1 for any coupling and a dynamic ESQPT at $x_c$2 only in the superradiant phase. The singularity is model dependent: for the Tavis–Cummings model it appears as discontinuities in derivatives of the density of states, whereas in the Dicke model the superradiant ESQPT at $x_c$3 yields a logarithmic divergence in $x_c$4 (Bastarrachea-Magnani et al., 2013).

Spinor Bose gases and Raman-dressed spin-orbit-coupled condensates realize ESQPTs in a cold-atom context. In the ferromagnetic spin-1 condensate at zero magnetization, the mean-field energy landscape develops a saddle at $x_c$5, and the separatrix is a figure-eight trajectory whose period diverges at the ESQPT. In the Raman-dressed system, the same collective-spin structure supports an excited broken-axisymmetry phase that maps onto an Excited-Stripe phase, with stripe contrast serving as a direct order parameter (Feldmann et al., 2020, Cabedo et al., 2021).

The three-level LMG model enlarges the semiclassical phase space and produces multiple separatrices generated by several stationary points. In that setting, ESQPT signatures persist but become entangled with mixed regular–chaotic dynamics, so the notion of a single dominant separatrix no longer suffices (Mayorgas et al., 26 Feb 2026).

3. Eigenstate structure, phase characterization, and order parameters

A central ESQPT signature is the abrupt reorganization of eigenstates in symmetry-adapted bases. In the $x_c$6 vibron realization, any eigenstate can be expanded in the spherical $x_c$7 basis or in the deformed $x_c$8 basis. The participation ratio,

$x_c$9

the inverse participation ratio, and the Shannon entropy all show that for $\nabla H(x_c)=0$0 the eigenstates closest to $\nabla H(x_c)=0$1 are highly localized in the $\nabla H(x_c)=0$2 basis, with dominant weight on the spherical ground-state component $\nabla H(x_c)=0$3. In the $\nabla H(x_c)=0$4 basis the anomaly is much weaker and becomes visible only at larger $\nabla H(x_c)=0$5 (Santos et al., 2015).

The infinite-range Ising formulation of the LMG model exhibits the same mechanism in spin language. In the $\nabla H(x_c)=0$6 basis, the participation ratio develops a pronounced dip at $\nabla H(x_c)=0$7, while in the $\nabla H(x_c)=0$8 basis it shows a discontinuous jump across the critical energy. Simultaneously, energy-resolved magnetizations bifurcate: below the separatrix one finds two symmetric branches $\nabla H(x_c)=0$9 associated with parity-related states, whereas above the separatrix all eigenstates satisfy $E_c=H(x_c)$0. This provides an explicit phase-structural distinction between deformed and spherical excited-state sectors (Santos et al., 2016).

Because standard ground-state order parameters do not generally extend in a simple way to ESQPTs, several works formulate phase labels directly in the excited spectrum. In the spin-1 spinor condensate, the mean-field trajectories below and above the critical line differ topologically: one phase consists of a single loop encircling the axis, while the other consists of two disconnected loops. This leads to a topological winding number

$E_c=H(x_c)$1

with $E_c=H(x_c)$2, $E_c=H(x_c)$3, or $E_c=H(x_c)$4 in the three excited-state phases TF$E_c=H(x_c)$5, P$E_c=H(x_c)$6, and BA$E_c=H(x_c)$7 (Feldmann et al., 2020).

The Raman-dressed spin-orbit-coupled gas translates this topology into an experimentally accessible excited-state order parameter. The Excited-Stripe phase corresponds to the BA$E_c=H(x_c)$8 phase of the spinor model, and its instantaneous stripe amplitude is

$E_c=H(x_c)$9

so a nonzero time-averaged stripe contrast identifies the excited-state phase directly in real-space density modulations (Cabedo et al., 2021).

A more general formulation introduces an operator

$r$0

which has eigenvalues $r$1 and becomes a constant of motion only in one excited-state phase. In the Rabi and Dicke models, $r$2 explains the change from degenerate doublets to nondegenerate levels across the ESQPT and shows that below $r$3 equilibrium may depend on both energy and the sector of $r$4, whereas above $r$5 energy alone is thermodynamically relevant (Corps et al., 2021).

4. Nonequilibrium dynamics and dynamical diagnostics

ESQPTs strongly constrain nonequilibrium evolution after sudden parameter changes. In $r$6 bosonic models, the relevant dynamical quantities are the survival probability

$r$7

and the associated local density of states

$r$8

When the initial state is a $r$9-basis vector whose mean energy lies near $f$0, the LDOS is sharply localized around the separatrix and the time evolution becomes extremely slow. This slowdown is specific to initial states aligned with the spherical basis; analogous $f$1-basis initial states do not display an ESQPT-induced anomaly (Santos et al., 2015).

The same phenomenon appears in the infinite-range Ising model and persists across the $f$2 family with $f$3. The initial state $f$4 is mapped near the critical energy and decays exceedingly slowly, whereas more delocalized initial states evolve faster. For several such delocalized states, the LDOS takes the form

$f$5

so that

$f$6

and the long-time decay is algebraic, $f$7. This connects ESQPT dynamics in long-range systems to the single-excitation XX-chain envelope, while leaving intact the ESQPT-specific slowing for separatrix-localized states (Santos et al., 2016, Pérez-Bernal et al., 2016).

In the spin-1 condensate, the Loschmidt echo spectrum generalizes survival-probability diagnostics by resolving overlaps with excited states of the initial Hamiltonian,

$f$8

At the critical quench $f$9, both the time-evolved and long-time-averaged Loschmidt echo spectrum undergo a marked reorganization, and the associated energy distribution develops a pronounced feature at the ESQPT energy. Its variance and entropy change rapidly at the critical quench, providing a state-resolved dynamical detector (Niu et al., 2022).

The anharmonic LMG model exhibits two ESQPTs, one inherited from the standard LMG saddle and one induced by the anharmonic boundary mechanism. Under the tangent-method quench protocol, the survival probability drops to nearly zero and then fluctuates irregularly when the quench lands exactly on either critical energy. The time-averaged Loschmidt echo,

$(f-1)$0

shows local maxima at both ESQPTs, while suitable microcanonical OTOCs are nonzero below the first ESQPT, vanish in the intermediate region, and become nonzero again above the second one. This establishes a dynamical nonzero/zero/nonzero pattern across the two separatrices (Khalouf-Rivera et al., 2022).

Microcanonical OTOCs also distinguish how symmetry-sector degeneracies affect ESQPT dynamics in two-level bosonic models. In the $(f-1)$1 case, gaps between opposite-parity sectors vanish exponentially with $(f-1)$2, and the infinite-time average of a parity-flipping OTOC is nonzero below $(f-1)$3 and zero above it, acting as an ESQPT order parameter already at finite size. For $(f-1)$4, the relevant inter-sector gaps close only algebraically, so the infinite-time average vanishes at finite $(f-1)$5 and acquires order-parameter status only in the mean-field limit (Khalouf-Rivera et al., 2023).

The entropy of the work distribution provides a further nonequilibrium diagnostic. In sudden quenches of the anharmonic LMG model, the Shannon entropy

$(f-1)$6

peaks near the critical quench strength for both ESQPTs. Its maximal value scales logarithmically, $(f-1)$7, with representative fitted $(f-1)$8 values near $(f-1)$9, while the shift of the maximum from criticality scales as $f$00 with representative $f$01 values around $f$02–$f$03 (Zhang et al., 2023).

5. Chaos, composed systems, non-Hermitian extensions, and constraints

A common misconception is that a density-of-states anomaly by itself is always sufficient to identify an ESQPT. In systems with mixed regular and chaotic dynamics, or in composed systems with multiple symmetry sectors, the evidence can be blurred and must be cross-validated by structural and dynamical diagnostics. The three-level LMG model makes this explicit: multiple separatrices $f$04 coexist with chaotic and quasi-integrable windows, so reliable identification requires combining mean-field separatrices, Peres lattices, participation ratios, Poincaré sections, nearest-neighbor spacing distributions, the degree of chaos $f$05, and the Kullback–Leibler divergence. In that setting, the Kullback–Leibler divergence is found to be a cleaner qualitative chaos-sensitive indicator than $f$06 (Mayorgas et al., 26 Feb 2026).

The same caution applies to composed algebraic systems. In the two-fluid Lipkin model, the ESQPT critical energy is fixed at $f$07 throughout the deformed region because it corresponds to the potential maximum at zero deformation, but the density of states can be misleading when different $f$08 symmetry families are superposed. Peres lattices and participation ratios locate the ESQPT much more reliably and also expose the excited-state phases below and above $f$09 (Garcia-Ramos et al., 2017). In the interacting boson model with two effective degrees of freedom, additional nonanalyticities arise from nontrivial kinetic stationary points and from the compact phase-space boundary, complicating any purely potential-based interpretation (1903.16551).

A non-Hermitian reformulation shows that ESQPT precursors in finite systems are governed by exceptional points. In the complexified LMG Hamiltonian, the avoided crossings that characterize the Hermitian ESQPT become exact EP crossings at complex values of the control parameter. As $f$10, the corresponding EP sequences approach the real axis and the separatrix; Padé extrapolation of the complex EP positions yields $f$11, with faster convergence for EPs nearest the ESQPT (Šindelka et al., 2016).

Floquet systems reveal a related but distinct viewpoint. In the quantum standard map, the ESQPT precursor is organized by the separatrix of an unstable periodic orbit, and the local quasienergy spacing behaves as

$f$12

so the local density diverges logarithmically in the semiclassical limit. As chaos increases, destructive interference between principal homoclinic orbits destroys the finite-size precursor when $f$13, showing that classical chaos does not immediately erase ESQPT signatures but suppresses them once the quantum resolution is sufficient to resolve the homoclinic structure (García-Mata et al., 2021).

Constrained systems generalize the semiclassical theory further. For algebraic boson models with fixed excitation number and additional integrals of motion, the effective number of degrees of freedom is reduced, which changes both the singular derivative order and the set of admissible stationary points. In fully connected bosonic systems, a single Holstein–Primakoff chart can miss ESQPT stationary points that lie on the boundary of the compact reduced phase space; a complete atlas of HP mappings is required to recover all ESQPTs. The Lagrange-multiplier construction avoids this coordinate singularity and recovers the full stationary-point structure directly on the constrained manifold (Novotný et al., 2024).

6. Finite-size behavior, experimental platforms, and open problems

Finite systems smooth all ESQPT singularities, but the characteristic precursors are systematic. Across the $f$14 pairing models, the infinite-range Ising model, and the general review framework, the density-of-states peaks sharpen with $f$15, participation-ratio dips deepen, avoided crossings narrow, and critical dynamical anomalies remain visible at sizes accessible to exact diagonalization or current simulators (Santos et al., 2015, Santos et al., 2016, Cejnar et al., 2020).

Several experimental platforms are already aligned with the diagnostics emphasized in the literature. Trapped-ion simulators realize long-range Ising models close to the LMG limit and can prepare the low-excitation states whose post-quench dynamics becomes anomalously slow near $f$16 (Pérez-Bernal et al., 2016). The same general family of models is proposed for trapped-ion observation of slow survival-probability decay and LDOS narrowing in the $f$17 setting (Santos et al., 2015). Spin-1 condensates provide two complementary routes: interferometric measurement of the topological order parameter through the population observable associated with $f$18 in the spinor case, and spectroscopy of the Loschmidt echo spectrum in the single-mode spinor gas (Feldmann et al., 2020, Niu et al., 2022). Raman-dressed spin-orbit-coupled gases offer an in situ density-based probe because the excited-state phase is encoded directly in stripe contrast (Cabedo et al., 2021). More broadly, cavity and circuit QED, molecular spectra, optical lattices, and photonic or driven systems are all identified as ESQPT-relevant arenas (Cejnar et al., 2020).

Open questions are consistent across the present literature. They include the universality of localization signatures across different ESQPT classes, precise scaling laws for participation-ratio dips, DOS peaks, and dynamical timescales, the extension to first-order ground-state transitions and coupled systems, the role of boundary and degenerate stationary points in higher-dimensional models, and the behavior of ESQPTs in nonintegrable, driven, dissipative, or strongly chaotic settings (Santos et al., 2015, Santos et al., 2016, Cejnar et al., 2020, Novotný et al., 2024). The existing results suggest that ESQPTs are best understood not as isolated density-of-states anomalies, but as global reorganizations of spectral geometry, eigenstate topology, and nonequilibrium response across the excited spectrum.